Eyes wrong position after import
The issue is in your skinning.

For a mesh to be deformed by a bone, you need this mesh to be assigned to a vertex group named exactly like the bone.

Your eyes are assigned to a single vertex group named "Eyes", while you have two bones, named "Eye.L" and "Eye.R". So you have to remove that "Eyes" group, create new groups with the bones's names and assign the geometry accordingly.

Also, for each bone not supposed to deform the mesh, you should uncheck the Deform panel:



This will avoid unnecessary calculations, memory overload, and such non-deforming bones won't create vertex groups as they don't need to.